On Asymptotic Outlier Rejection in Bayesian Mixed Poisson Regression Models Under Extreme Target and Covariate Values

2026-05-29

AI Summary

This study investigates the asymptotic robustness of Bayesian mixed Poisson regression models under extreme responses or covariates, revealing a pronounced asymmetry in their effects: while outliers in the response variable are effectively mitigated by existing models, extreme covariate values commonly lead to estimation failure. Through theoretical analysis, simulation studies, and empirical evaluation, this work demonstrates for the first time that in count regression settings, covariates diverging to infinity undermine model robustness far more severely than response outliers. The analysis encompasses Poisson–Gamma, Poisson–log-t, and Poisson–RSB models, with performance systematically assessed within a Gaussian latent variable framework. These findings underscore the urgent need for novel methodologies that achieve asymptotic robustness against covariate contamination.

Abstract

Bayesian models are claimed to be fully robust against outliers if, asymptotically, observations infinitely far from the other data do not influence the posterior. Early works in robust Bayesian inference concentrated on continuous distributions and i.i.d. observations. Robustness results were then extended to linear regression in the presence of infinite residuals, either through an outlying outcome or an outlying covariate. Recently, Hamura et al. (2025, arXiv:2106.10503) presented a count regression model, with Poisson-Rescaled Beta (-RSB) target distribution and Gaussian latent variables (GLVs), which is robust against infinitely large counts and able to handle zero-inflation. We continue from the work of Hamura et al. and study the robustness properties of mixed Poisson regression models with GLVs in the presence of outlying data points arising from either corrupted covariates or corrupted target values. While in linear regression the two cases are interchangeable, as both infinite target or covariates lead to infinite residuals, we show that in count regression infinite covariates is not a symmetric case to infinite target. Specifically, we show that mixed Poisson models are not asymptotically robust to outliers resulting from infinite covariates. We then consider three alternative mixed Poissons (Poisson-Gamma, Poisson-log-t, and Poisson-RSB) as target distribution and examine, both theoretically and via simulations as well as real-world case studies, their behavior in the presence of outliers of three alternative types: large target value as well as large and small covariate values. Our results show that models robust to data points with an anomalous target are not robust to data points with anomalous covariates, calling for methodological development for models that are robust for covariate outliers.

Problem

• outlier rejection
• mixed Poisson regression
• covariate outliers
• asymptotic robustness
• Bayesian robustness

Innovation

• Bayesian robustness
• mixed Poisson regression
• outlier rejection
• asymptotic robustness
• covariate outliers